Are proofs equivalent to dovetailing computations?

[Jason Resch]

In "The Universal Numbers. From Biology to Physics" Bruno writes

"The universal dovetailing can be seen as the proofs of all true Sigma_1 propositions there exists x,y,z such that P_x(y) = z, with some sequences of such propositions mimicking the infinite failing or proving some false Sigma_1 propositions."

This is something I was thinking about recently in the context of universal Diophantine equations. It seems more correct to me to say these equations don't themselves represent the execution traces of the programs, but rather represent proofs of the outputs of programs.

This can be seen from the fact that the work of verifying a Diophantine equation requires only a finite and constant number of arithmetical operations, while the computation itself could involve much more work, in terms of arithmetical steps.

So is it right to say that the proof of the result of some computable function is different from the computable function itself?  In other words, a fixed Diophantine equation, regardless of the values of its variables, does not itself yield conscious mind states, though it points to the existence of another object in math (the universal machine) whose operation would yield the conscious mind states?

I am just trying to develop a more clear picture in my mind of the relation between arithmetic, proofs, computational traces, and mind states.

Jason

[Jason Resch]

Another comment/question:

"with some sequences of such propositions mimicking the infinite failing or proving some false Sigma_1 propositions." 

Assuming the undecidability of the Mandelbrot set, does this undedicability have the same implications as the above "mimicking the infinite failing or proving some false Sigma_1 propositions"?  In a poetic manner of speaking, does the shape of the Mandelbrot set's edge somehow mirror the shape of non-computable functions in the UD?

Can we infer anything about the existence of all computations from the mathematical existence of the complete Mandelbrot set alone?

Jason

[Bruno Marchal]

Jason,

Interesting and important questions. Unfortunately today I have family duties … ,

 I will answer in the evening or tomorrow. (Same for possible other posts),

Best,

Bruno

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[Philip Thrift]

If only there were a dovetailer to multiplex all one's duties. :)

@philipthrift

[Russell Standish]

On Tue, Aug 13, 2019 at 02:41:09AM -0700, Philip Thrift wrote:
>
> If only there were a dovetailer to multiplex all one's duties. :)

They made a movie about that, starring Tom Hanks IIRC. Can't tremeber
the title, though...


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----------------------------------------------------------------------------
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Principal, High Performance Coders
Visiting Senior Research Fellow hpc...@hpcoders.com.au
Economics, Kingston University http://www.hpcoders.com.au
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[Bruno Marchal]

On 12 Aug 2019, at 23:36, Jason Resch <jason...@gmail.com> wrote:

In "The Universal Numbers. From Biology to Physics" Bruno writes

"The universal dovetailing can be seen as the proofs of all true Sigma_1 propositions there exists x,y,z such that P_x(y) = z, with some sequences of such propositions mimicking the infinite failing or proving some false Sigma_1 propositions."

This is something I was thinking about recently in the context of universal Diophantine equations. It seems more correct to me to say these equations don't themselves represent the execution traces of the programs, but rather represent proofs of the outputs of programs.

This can be seen from the fact that the work of verifying a Diophantine equation requires only a finite and constant number of arithmetical operations, while the computation itself could involve much more work, in terms of arithmetical steps.

We have to distinguish

- a computable function (from N to N, say), That is an infinite object, which can be represented by an infinite set (the set of input-output). That set is often called the “graph” of the function. We can show that if the function is computable, its graph is mechanically generable (recursively enumerable). Such a set can be described by a sigma_1 sentence (that is a sentence having the shape {y such that ExP(x, y)}.

- The code of a computable function. That is a finite object (representable using some word or numbers, or finite set, …). I use usually the word “digital machine” for this. It is the (virtual) body of the machine. It is finitely describable.

- a computation. That is either a finite or an infinite thing, usually representable by a sequence of numbers or words, called the trace of the computation). Some author call “computation” only the computations which halt, and thus are finite, like Martin Davis in the book “Computability and Unsolvability). Other like Daniel E. Cohen, and me most of the time, admits the term “computation” for non halting, and thus infinite, one. 

With, this you can guess that when you have an halting computation, you have automatically a proof of a sigma_1 sentence (like the sentence saying that some input-output (x,y) belongs to the graph of that function, or that x belongs to its domain).

Unfortunately, the reciprocal is not necessarily true. We can find a proof of a sigma_1 sentence which would not be a computation, like an indirect proof by a reduction ad absurdum which can be a non constructive proof of an existential statement. Such a proof would state that a computation is halting, without executing that computation.

So, proofs (of sigma_1 sentence) are not necessarily a computation, and as such will not belong to the universal dovetailing directly, although the universal dovetailing will emulate some subjects doing those non-constructive proof, but that gives a computation emulating a reasoning process, and that reasoning process does not need to be a computation.

Inversely, as I said, a computation can be seen as a proof of a sigma_1 sentence. There is a theorem which makes this precise: the normal form theorem of Kleene, which shows that there are computable (and elementary, primitive recursive, if people remember the definition) U and T such that if f(x) is a computable function, computed by the code z, then 

 y = f(x) = U(the least c (T(z, x, c)).

T(z, x, c) is Kleene’s predicate. It says that z is the Gödel number of a (partial) computable function, x is the input given to the machine with code z, and y is an halting computation given by the machine with code z when applied to x.

U is the result extracting function, which gives the output y from the computation c. Usually U just take the last line of the computation, for example. This results shows that you can always program a computable function with one “while”, or that you can always well structure a code (suppress all the “goto”-like instruction by just one “while”.

So is it right to say that the proof of the result of some computable function is different from the computable function itself? 

So, the proof is different from the computable function, and from its code. But the computation (if it halts on some input) will provide gives a proof that x belongs to its domain, or a proof that (x, y) belongs to its graph, if y = U(c) like above.

Note that if a proof is sufficiently constructive, it will be (equivalent) to a computation.

Constructive proof of sigma_1 sentences can be seen as equivalent to halting computation. 

Halting computation can be seen as equivalent to (constructive) proof.

In other words, a fixed Diophantine equation, regardless of the values of its variables, does not itself yield conscious mind states, though it points to the existence of another object in math (the universal machine) whose operation would yield the conscious mind states?

Diophantine polynomial are Turing universal, so there is a specific Diophantine polynomial which is universal.

There is a universal Diophantine polynomial equation, like there is universal Turing machine, a universal combinators, a universal lambda expression, a universal game of life pattern, etc.  Now, such an equation (combinators, machines, pattern) can be seen as a code. It is a finite static thing, and there is no consciousness associated with it per se, but there will be a consciousness associated to the verifying process if that polynomial equation as this or that particular number as solution.

I am just trying to develop a more clear picture in my mind of the relation between arithmetic, proofs, computational traces, and mind states.

Yes, it can be confusing, not mentioning the confusion between a mathematical object or statement and their syntactical description. That is even more confusing for the notion of (halting) computation, which is a finite object, yet still different from all its finite description. The computation is in the relation (eventually definable in arithmetic) between token (number) which makes that description describing a computation (which is an abstract relation).

You ask in a sequel:

Another comment/question:

"with some sequences of such propositions mimicking the infinite failing or proving some false Sigma_1 propositions." 

Assuming the undecidability of the Mandelbrot set, does this undedicability have the same implications as the above "mimicking the infinite failing or proving some false Sigma_1 propositions"?  In a poetic manner of speaking, does the shape of the Mandelbrot set's edge somehow mirror the shape of non-computable functions in the UD?

That is plausible. Unfortunately, to my knowledge, this is still an open problem for the Rational Mandelbrot set (the problem of deciding if (a+bi) with a and b rational belongs to the Mandelbrot set.

For the “Real” Mandelbrot set (with a and b real) the corresponding problem has been solved for a special notion of computability with respect to a ring (or a field), by Blum, Smale and Shub(*). But of course, in that case a+bi might be an infinite object, and does not represent programs (there is no “Church-Thesis” for the notion of computability with real numbers: there are many non equivalent definitions). The work of Blum & Al. is more interesting for Numerical Analysis than for Number theology or Digital Mechanist Metaphysics.

Can we infer anything about the existence of all computations from the mathematical existence of the complete Mandelbrot set alone?

Yes, but only trivially. The Mandelbrot assumes the Complex Plane, which is Turing universal, because it is an extension of first order analysis + a mean to define the natural numbers and their addition and muliplication. laws.

But if the conjecture above, on the rational Mandelbrot set, is true, and if the undecidability is of the usual type (creative set, not simple set, in the sense of Emil Post), then the Mandelbrot set becomes a compact description of the universal dovetailing, or of any UD* (the complete running of some UD). That would be nice, but that is still an open problem. Solving the case on the real does not help solving the case on the natural numbers.

Bruno

(*) Blum L., Cucker F., Shub M., Smale S. Complexity and Real Computation, 1998, Springer Verlag, New-York.

[Jason Resch]

On Wed, Aug 14, 2019 at 5:02 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 12 Aug 2019, at 23:36, Jason Resch <jason...@gmail.com> wrote:

In "The Universal Numbers. From Biology to Physics" Bruno writes

"The universal dovetailing can be seen as the proofs of all true Sigma_1 propositions there exists x,y,z such that P_x(y) = z, with some sequences of such propositions mimicking the infinite failing or proving some false Sigma_1 propositions."

This is something I was thinking about recently in the context of universal Diophantine equations. It seems more correct to me to say these equations don't themselves represent the execution traces of the programs, but rather represent proofs of the outputs of programs.

This can be seen from the fact that the work of verifying a Diophantine equation requires only a finite and constant number of arithmetical operations, while the computation itself could involve much more work, in terms of arithmetical steps.

We have to distinguish

- a computable function (from N to N, say), That is an infinite object, which can be represented by an infinite set (the set of input-output). That set is often called the “graph” of the function. We can show that if the function is computable, its graph is mechanically generable (recursively enumerable). Such a set can be described by a sigma_1 sentence (that is a sentence having the shape {y such that ExP(x, y)}.

Does the identity between a computation (in terms of discrete steps with counterfactual behavior), and its representation as the set of all inputs to outputs imply that the Blockhead (giant state table brain) possesses consciousness of the same form as the incrementally processed computation?  Perhaps I just have too myopic of a view of what computation is from my familiarity with programming.  

 

- The code of a computable function. That is a finite object (representable using some word or numbers, or finite set, …). I use usually the word “digital machine” for this. It is the (virtual) body of the machine. It is finitely describable.

- a computation. That is either a finite or an infinite thing, usually representable by a sequence of numbers or words, called the trace of the computation). Some author call “computation” only the computations which halt, and thus are finite, like Martin Davis in the book “Computability and Unsolvability). Other like Daniel E. Cohen, and me most of the time, admits the term “computation” for non halting, and thus infinite, one. 

With, this you can guess that when you have an halting computation, you have automatically a proof of a sigma_1 sentence (like the sentence saying that some input-output (x,y) belongs to the graph of that function, or that x belongs to its domain).

Unfortunately, the reciprocal is not necessarily true. We can find a proof of a sigma_1 sentence which would not be a computation, like an indirect proof by a reduction ad absurdum which can be a non constructive proof of an existential statement. Such a proof would state that a computation is halting, without executing that computation.

So, proofs (of sigma_1 sentence) are not necessarily a computation, and as such will not belong to the universal dovetailing directly, although the universal dovetailing will emulate some subjects doing those non-constructive proof, but that gives a computation emulating a reasoning process, and that reasoning process does not need to be a computation.

Inversely, as I said, a computation can be seen as a proof of a sigma_1 sentence. There is a theorem which makes this precise: the normal form theorem of Kleene, which shows that there are computable (and elementary, primitive recursive, if people remember the definition) U and T such that if f(x) is a computable function, computed by the code z, then 

 y = f(x) = U(the least c (T(z, x, c)).

T(z, x, c) is Kleene’s predicate. It says that z is the Gödel number of a (partial) computable function, x is the input given to the machine with code z, and y is an halting computation given by the machine with code z when applied to x.

U is the result extracting function, which gives the output y from the computation c. Usually U just take the last line of the computation, for example. This results shows that you can always program a computable function with one “while”, or that you can always well structure a code (suppress all the “goto”-like instruction by just one “while”.

I think this fact (of implementing any program with a single loop) explains both why recursive functions are Turing complete and why CPUs can work by simply repeating the application of some electrical circuit over and over again.

 

So is it right to say that the proof of the result of some computable function is different from the computable function itself? 

So, the proof is different from the computable function, and from its code. But the computation (if it halts on some input) will provide gives a proof that x belongs to its domain, or a proof that (x, y) belongs to its graph, if y = U(c) like above.

Note that if a proof is sufficiently constructive, it will be (equivalent) to a computation.

Constructive proof of sigma_1 sentences can be seen as equivalent to halting computation. 

Halting computation can be seen as equivalent to (constructive) proof.

In other words, a fixed Diophantine equation, regardless of the values of its variables, does not itself yield conscious mind states, though it points to the existence of another object in math (the universal machine) whose operation would yield the conscious mind states?

Diophantine polynomial are Turing universal, so there is a specific Diophantine polynomial which is universal.

There is a universal Diophantine polynomial equation, like there is universal Turing machine, a universal combinators, a universal lambda expression, a universal game of life pattern, etc.  Now, such an equation (combinators, machines, pattern) can be seen as a code. It is a finite static thing, and there is no consciousness associated with it per se, but there will be a consciousness associated to the verifying process if that polynomial equation as this or that particular number as solution.

While I agree and understand that universal Diophantine polynomial solutions have the same form as any computable set, is the Diophantine equation more correctly viewed a computation, or as a proof of correctness for some computation?  In the other thread you mentioned the fact that a nested exponentiation of 1000 terms can be verified in 100 additions and multiplications.  Are these 100 additions and multiplications truly a sufficient reenactment of the computational relation in the same form necessary for consciousness (is it the arithmetical truth of the relation that is conscious, or the processing of information by the computation that is conscious), is there really no important difference between the Diophantine proof and the constructive proof generated by the operation of a Turing machine?

(Note, I first learned of and was blown away by the possibility of verifying any computation in constant time, as it pertains to practical applications in the form of "zk-SNARKs" (zero knowledge, succinct non-interactive argument of knowledge) https://www.youtube.com/watch?v=nS3smRAfUd8  https://eprint.iacr.org/2013/507.pdf they have numerous practical applications but are computationally difficult to create (but easy to check)).

 

I am just trying to develop a more clear picture in my mind of the relation between arithmetic, proofs, computational traces, and mind states.

Yes, it can be confusing, not mentioning the confusion between a mathematical object or statement and their syntactical description. That is even more confusing for the notion of (halting) computation, which is a finite object, yet still different from all its finite description. The computation is in the relation (eventually definable in arithmetic) between token (number) which makes that description describing a computation (which is an abstract relation).

You ask in a sequel:

Another comment/question:

"with some sequences of such propositions mimicking the infinite failing or proving some false Sigma_1 propositions." 

Assuming the undecidability of the Mandelbrot set, does this undedicability have the same implications as the above "mimicking the infinite failing or proving some false Sigma_1 propositions"?  In a poetic manner of speaking, does the shape of the Mandelbrot set's edge somehow mirror the shape of non-computable functions in the UD?

That is plausible. Unfortunately, to my knowledge, this is still an open problem for the Rational Mandelbrot set (the problem of deciding if (a+bi) with a and b rational belongs to the Mandelbrot set.

For the “Real” Mandelbrot set (with a and b real) the corresponding problem has been solved for a special notion of computability with respect to a ring (or a field), by Blum, Smale and Shub(*). But of course, in that case a+bi might be an infinite object, and does not represent programs (there is no “Church-Thesis” for the notion of computability with real numbers: there are many non equivalent definitions). The work of Blum & Al. is more interesting for Numerical Analysis than for Number theology or Digital Mechanist Metaphysics.

Can we infer anything about the existence of all computations from the mathematical existence of the complete Mandelbrot set alone?

Yes, but only trivially. The Mandelbrot assumes the Complex Plane, which is Turing universal, because it is an extension of first order analysis + a mean to define the natural numbers and their addition and muliplication. laws.

But if the conjecture above, on the rational Mandelbrot set, is true, and if the undecidability is of the usual type (creative set, not simple set, in the sense of Emil Post), then the Mandelbrot set becomes a compact description of the universal dovetailing, or of any UD* (the complete running of some UD). That would be nice, but that is still an open problem. Solving the case on the real does not help solving the case on the natural numbers.

Thanks for the background and explanation.  Is it the case then that any undecidable (creative?) set is a compact description of universal dovetailing?  Would Chaitin's constant also qualify as a compact description of the universal dovetailing (though being a single real number, rather than a set of rational complex points)?

Jason

[Russell Standish]

On Fri, Aug 16, 2019 at 12:06:32PM -0500, Jason Resch wrote:
>
> Thanks for the background and explanation.  Is it the case then that any
> undecidable (creative?) set is a compact description of universal dovetailing? 
> Would Chaitin's constant also qualify as a compact description of the universal
> dovetailing (though being a single real number, rather than a set of rational
> complex points)?
>

Related to this, on page 218 of Li and Vitanyi's "Introduction to Kolmogorov Complexity and it Applications", right under corollary 3.6.2 is the statement:

"Moreover, for all axiomatic mathematical theories that can be
extressed compactly enough to be conceivably interesting to human
beings, say in fewer than 10,000 bits, [the first 10,000 bits of the
Chatin probability Ω] can be used to decide for every statement in the
theory whether it is the true, false or independent. ... Thus Ω is
truly the number of Wisdom, and 'can be known of, but not known,
through human reason' [C.H Bennett and M. Gardner, Sci
Am. 241:11(1979),20-34]".

Cheers

[Jason Resch]

On Fri, Aug 16, 2019 at 6:31 PM Russell Standish <li...@hpcoders.com.au> wrote:

On Fri, Aug 16, 2019 at 12:06:32PM -0500, Jason Resch wrote:
>
> Thanks for the background and explanation.  Is it the case then that any
> undecidable (creative?) set is a compact description of universal dovetailing? 
> Would Chaitin's constant also qualify as a compact description of the universal
> dovetailing (though being a single real number, rather than a set of rational
> complex points)?
>

Related to this, on page 218 of Li and Vitanyi's "Introduction to Kolmogorov Complexity and it Applications", right under corollary 3.6.2 is the statement:

"Moreover, for all axiomatic mathematical theories that can be
extressed compactly enough to be conceivably interesting to human
beings, say in fewer than 10,000 bits, [the first 10,000 bits of the
Chatin probability Ω] can be used to decide for every statement in the
theory whether it is the true, false or independent. ... Thus Ω is
truly the number of Wisdom, and 'can be known of, but not known,
through human reason' [C.H Bennett and M. Gardner, Sci
Am. 241:11(1979),20-34]".

What an incredible and fascinating discovery/insight.  I couldn't understand it at first but did some searching and reading and came across a detailed explanation.  The quote above seems to be based on an earlier work by Charles Bennett, in "On Random and Hard-to-Describe Numbers" ( http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.214.7866&rep=rep1&type=pdf ).  In it I found another nice quote:

Throughout history philosophers and mystics have sought a compact key to universal wisdom, a finite formula or text which, when known and understood, would provide the answer to every question. The Bible, the Koran, the mythical secret books of Hermes Trismegistus, and the medieval Jewish Cabala have been so regarded. Sources of universal wisdom are traditionally protected from casual use by being hard to find, hard to understand when found, and dangerous to use, tending to answer more and deeper questions than the user wishes to ask. Like God the esoteric book is simple yet undescribable, omniscient, and transforms all who know It. The use of classical texts to foretell mundane events is considered superstitious nowadays, yet, in another sense, science is in quest of its own Cabala, a concise set of natural laws which would explain all phenomena. In mathematics, where no set of axioms can hope to prove all true statements, the goal might be a concise axiomatization of all “interesting” true statements.

Ω is in many senses a Cabalistic number. It can be known of, but not known, through human reason. To know it in detail, one would have to accept its uncomputable digit sequence on faith, like words of a sacred text. It embodies an enormous amount of wisdom in a very small space, inasmuch as its first few thousand digits, which could be written on a small piece of paper, contain the answers to more mathematical questions than could be written down in the entire universe, including all interesting finitely refutable conjectures. Its wisdom is useless precisely because it is universal: the only known way of extracting from Ω the solution to one halting problem, say the Fermat conjecture, is by embarking on a vast computation that would at the same time yield solutions to all other equally simply-stated halting problems, a computation far too large to be carried out in practice. Ironically, although Ω cannot be computed, it might accidentally be generated by a random process, e.g. a series of coin tosses, or an avalanche that left its digits spelled out in the pattern of boulders on a mountainside. The initial few digits of Ω are thus probably already recorded somewhere in the universe. Unfortunately, no mortal discoverer of this treasure could verify its authenticity or make practical use of it. 

Jason

[Philip Thrift]

Chaitin numbers of course can also be extended to the computational hierarchy (oracles, Turing jump iterations of the halting problem):

Super-Ω

https://en.wikipedia.org/wiki/Chaitin%27s_constant#Super_Omega

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2011.0319

@philipthrift

[Bruno Marchal]

On 16 Aug 2019, at 19:06, Jason Resch <jason...@gmail.com> wrote:

On Wed, Aug 14, 2019 at 5:02 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 12 Aug 2019, at 23:36, Jason Resch <jason...@gmail.com> wrote:

In "The Universal Numbers. From Biology to Physics" Bruno writes

"The universal dovetailing can be seen as the proofs of all true Sigma_1 propositions there exists x,y,z such that P_x(y) = z, with some sequences of such propositions mimicking the infinite failing or proving some false Sigma_1 propositions."

This is something I was thinking about recently in the context of universal Diophantine equations. It seems more correct to me to say these equations don't themselves represent the execution traces of the programs, but rather represent proofs of the outputs of programs.

This can be seen from the fact that the work of verifying a Diophantine equation requires only a finite and constant number of arithmetical operations, while the computation itself could involve much more work, in terms of arithmetical steps.

We have to distinguish

- a computable function (from N to N, say), That is an infinite object, which can be represented by an infinite set (the set of input-output). That set is often called the “graph” of the function. We can show that if the function is computable, its graph is mechanically generable (recursively enumerable). Such a set can be described by a sigma_1 sentence (that is a sentence having the shape {y such that ExP(x, y)}.

Does the identity between a computation (in terms of discrete steps with counterfactual behavior), and its representation as the set of all inputs to outputs imply that the Blockhead (giant state table brain) possesses consciousness of the same form as the incrementally processed computation?  Perhaps I just have too myopic of a view of what computation is from my familiarity with programming.  

You cannot identify a computation and a representation of that computation. So the answer is no: the blockhead or the infinite look-up table does not process a computation.

Now, if in a theory, say ZF, you can define that representation of computation and prove that it represents a computation, then it in the model at that theory, it might make sense to say that it proves the existence of a computation, but the computations cannot be identified with its representation, even if the proof of the existence of that representation can be used to prove the existence of a computation, but only if that is proved itself in the theory (and that the theory is consistent, so that it has a model, where the computation will exist or be realised).

 

- The code of a computable function. That is a finite object (representable using some word or numbers, or finite set, …). I use usually the word “digital machine” for this. It is the (virtual) body of the machine. It is finitely describable.

- a computation. That is either a finite or an infinite thing, usually representable by a sequence of numbers or words, called the trace of the computation). Some author call “computation” only the computations which halt, and thus are finite, like Martin Davis in the book “Computability and Unsolvability). Other like Daniel E. Cohen, and me most of the time, admits the term “computation” for non halting, and thus infinite, one. 

With, this you can guess that when you have an halting computation, you have automatically a proof of a sigma_1 sentence (like the sentence saying that some input-output (x,y) belongs to the graph of that function, or that x belongs to its domain).

Unfortunately, the reciprocal is not necessarily true. We can find a proof of a sigma_1 sentence which would not be a computation, like an indirect proof by a reduction ad absurdum which can be a non constructive proof of an existential statement. Such a proof would state that a computation is halting, without executing that computation.

So, proofs (of sigma_1 sentence) are not necessarily a computation, and as such will not belong to the universal dovetailing directly, although the universal dovetailing will emulate some subjects doing those non-constructive proof, but that gives a computation emulating a reasoning process, and that reasoning process does not need to be a computation.

Inversely, as I said, a computation can be seen as a proof of a sigma_1 sentence. There is a theorem which makes this precise: the normal form theorem of Kleene, which shows that there are computable (and elementary, primitive recursive, if people remember the definition) U and T such that if f(x) is a computable function, computed by the code z, then 

 y = f(x) = U(the least c (T(z, x, c)).

T(z, x, c) is Kleene’s predicate. It says that z is the Gödel number of a (partial) computable function, x is the input given to the machine with code z, and y is an halting computation given by the machine with code z when applied to x.

U is the result extracting function, which gives the output y from the computation c. Usually U just take the last line of the computation, for example. This results shows that you can always program a computable function with one “while”, or that you can always well structure a code (suppress all the “goto”-like instruction by just one “while”.

I think this fact (of implementing any program with a single loop) explains both why recursive functions are Turing complete and why CPUs can work by simply repeating the application of some electrical circuit over and over again.

The enumeration of all partial recursive functions is Turing complete/univeral (by definition), and one partial recursive function (the universal one) is Turing complete/universal, also by definition. (The term “complete” is reserved for theories, and “universal” is used for function or machine). Now, some CPU might implements more than one loop, for reason of efficiency. 

 

So is it right to say that the proof of the result of some computable function is different from the computable function itself? 

So, the proof is different from the computable function, and from its code. But the computation (if it halts on some input) will provide gives a proof that x belongs to its domain, or a proof that (x, y) belongs to its graph, if y = U(c) like above.

Note that if a proof is sufficiently constructive, it will be (equivalent) to a computation.

Constructive proof of sigma_1 sentences can be seen as equivalent to halting computation. 

Halting computation can be seen as equivalent to (constructive) proof.

In other words, a fixed Diophantine equation, regardless of the values of its variables, does not itself yield conscious mind states, though it points to the existence of another object in math (the universal machine) whose operation would yield the conscious mind states?

Diophantine polynomial are Turing universal, so there is a specific Diophantine polynomial which is universal.

There is a universal Diophantine polynomial equation, like there is universal Turing machine, a universal combinators, a universal lambda expression, a universal game of life pattern, etc.  Now, such an equation (combinators, machines, pattern) can be seen as a code. It is a finite static thing, and there is no consciousness associated with it per se, but there will be a consciousness associated to the verifying process if that polynomial equation as this or that particular number as solution.

While I agree and understand that universal Diophantine polynomial solutions have the same form as any computable set, is the Diophantine equation more correctly viewed a computation, or as a proof of correctness for some computation? 

In this case, the computation is either in the search of a solution of a (non universal) polynomial Diophantine equation, or in the solution itself of one particular *universal* Diophantine equation.

The computation is always in the model, interpretation, or true relation, not in the number relation’s description in a theory. 

Keep in mind that a computation is an abstract *true* relation between numbers, or physical pieces, or anything which we accept as existing (and that depends on the base theory which is chosen). Now, by universality, we can chose any Turing complete theory, but once it has been chosen, we might need to carefully distinguish the level of “reality”. The universal diophantine polynomial equation is a finite object/program. Searching for all its solution (by a human, or by a game-of-life pattern in some bract space) will be equivalent with the running of a universal dovetailer. 

It is unfortunately easier to confuse a computation with one of its description, that a number with one of its description, but it is the same type of confusion, on a more sophisticated notion, though. I guess we will have opportunities to come back on this.

In the other thread you mentioned the fact that a nested exponentiation of 1000 terms can be verified in 100 additions and multiplications.  Are these 100 additions and multiplications truly a sufficient reenactment of the computational relation in the same form necessary for consciousness

Yes. The computation will be in the act of doing those addition and multiplication. You will need to do that already in some model of a Turing complete theory, be it a physical universe, or the arithmetical reality, etc.

(is it the arithmetical truth of the relation that is conscious, or the processing of information by the computation that is conscious),

It is in the processing, but the point, with the universal things, is that the processing is itself in the number relations, the one involving some universal system.

is there really no important difference between the Diophantine proof and the constructive proof generated by the operation of a Turing machine?

There is a difference of level. The true existence relatively to you will depend where you are emulated, a bit like in the explanation of how a virtual typhoon can make you wet. You need to be emulated at the right level. 

(Note, I first learned of and was blown away by the possibility of verifying any computation in constant time, as it pertains to practical applications in the form of "zk-SNARKs" (zero knowledge, succinct non-interactive argument of knowledge) https://www.youtube.com/watch?v=nS3smRAfUd8  https://eprint.iacr.org/2013/507.pdf they have numerous practical applications but are computationally difficult to create (but easy to check)).

Most of the time, verifying that something meet a specification is far easier than finding that something. But that is still relying on complex conjectures.

 

I am just trying to develop a more clear picture in my mind of the relation between arithmetic, proofs, computational traces, and mind states.

Yes, it can be confusing, not mentioning the confusion between a mathematical object or statement and their syntactical description. That is even more confusing for the notion of (halting) computation, which is a finite object, yet still different from all its finite description. The computation is in the relation (eventually definable in arithmetic) between token (number) which makes that description describing a computation (which is an abstract relation).

You ask in a sequel:

Another comment/question:

"with some sequences of such propositions mimicking the infinite failing or proving some false Sigma_1 propositions." 

Assuming the undecidability of the Mandelbrot set, does this undedicability have the same implications as the above "mimicking the infinite failing or proving some false Sigma_1 propositions"?  In a poetic manner of speaking, does the shape of the Mandelbrot set's edge somehow mirror the shape of non-computable functions in the UD?

That is plausible. Unfortunately, to my knowledge, this is still an open problem for the Rational Mandelbrot set (the problem of deciding if (a+bi) with a and b rational belongs to the Mandelbrot set.

For the “Real” Mandelbrot set (with a and b real) the corresponding problem has been solved for a special notion of computability with respect to a ring (or a field), by Blum, Smale and Shub(*). But of course, in that case a+bi might be an infinite object, and does not represent programs (there is no “Church-Thesis” for the notion of computability with real numbers: there are many non equivalent definitions). The work of Blum & Al. is more interesting for Numerical Analysis than for Number theology or Digital Mechanist Metaphysics.

Can we infer anything about the existence of all computations from the mathematical existence of the complete Mandelbrot set alone?

Yes, but only trivially. The Mandelbrot assumes the Complex Plane, which is Turing universal, because it is an extension of first order analysis + a mean to define the natural numbers and their addition and muliplication. laws.

But if the conjecture above, on the rational Mandelbrot set, is true, and if the undecidability is of the usual type (creative set, not simple set, in the sense of Emil Post), then the Mandelbrot set becomes a compact description of the universal dovetailing, or of any UD* (the complete running of some UD). That would be nice, but that is still an open problem. Solving the case on the real does not help solving the case on the natural numbers.

Thanks for the background and explanation.  Is it the case then that any undecidable (creative?) set is a compact description of universal dovetailing? 

By compact, I meant it in the usual topological-metrical sense. The Mandelbrot set is compact because you can enclose it is a circle. It is bounded geometrical object. It does not apply to set per se. But that is a detail.

Now, a set can be undecidable, but not Turing complete (creative). That has been found by Emil Post, with his notion of “simple set”, which are undecidable, but not complete.

The complement C of a complete set T is productive: C is constructively not recursively enumerable (RE). It has RE subset, and for each RE subset w_i included in C you can mechanically generate a counter-example: un element g(i) which is not inT, nor in w_i.

The complement C of a simple set S is immune (more complex than productive), it has no recursively enumerable subset at all. All attempt to build a w_i in C will cross (intersect) C soon or later. 

A creative set cannot explore the full universe, but can send more and more powerful rocket exploring it, and for each rocket  it can build a more powerful one, like we can extend PA or ZF in the transfinite.

A simple set can only build rockets falling back on the ground!

Would Chaitin's constant also qualify as a compact description of the universal dovetailing (though being a single real number, rather than a set of rational complex points)?

It does not. In fact Chaitin’s set (or “real number”) is not creative (Turing universal) but “simple", in that Post sense given above. You can’t compute anything with Chaitin’s number. It is like a box which contains all the gold of the universe, but there is no keys to open that box.

But “Post's number” , which decimals says if the nth program, in an enumeration of programs without inputs, stop or not, is creative, and “equivalent” with a UD* (seen properly in the right structure). But  the term ”compact” does not really apply here, unless perhaps you write the digits in smaller and smaller font so that you can write it all on one page.

You can look at Chaitin’s number as a maximal compression of Post’s number. Post number is deep, in Bennett sense, where Chaitin numbers is shallow and ultra-random. Chaitin’s number is the Post’s number with all the redundancies removed. You cannot do anything with it, except gives a non constructive proof of Gödel’s incompleteness (which was already in Emil Post, but without that “probability” interpretation of “simplicity”.

An interesting paper, that Brent points to me some years ago, which shows this and more is the paper by Calude and Hay: "Every Computably Enumerable Random Real Is Provably Computably Enumerable Random" (arXiv:0808.2220v5). Here: https://arxiv.org/abs/0808.2220

That paper is also useful to see that PA can prove the existence of universal numbers, computations, … (without assuming anything in physics, which could help some people here). But it is a bit technical. It also assume that ZF is arithmetically sound, which I believe, but is not that obvious, especially with Mechanism!

Both Chaitin and Post numbers contains all the secrets (of the universal dovetailing), but Chaitin’s number, by removing all the redundancies, is unreadable, and just as good as total randomness or mess. Post’s numbers on the contrary is comprehensible by all universal machines, so to speak.

Put in another way: with Post numbers, there is full hope for a decent measure on the relative computational histories. With Chaitin’s number, there is no measure at all, like if all computational histories were unique, somehow.

This does not mean that Chaitin’s number is not interesting for Mechanism. I think it will play some role in the thermodynamic of the computationalist physical reality, but not in its origin (which Post’s number does).

Bruno

Jason

 

Bruno

(*) Blum L., Cucker F., Shub M., Smale S. Complexity and Real Computation, 1998, Springer Verlag, New-York.

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[Russell Standish]

On Sat, Aug 17, 2019 at 12:17:38PM +0200, Bruno Marchal wrote:
>
> You cannot identify a computation and a representation of that computation. So
> the answer is no: the blockhead or the infinite look-up table does not process
> a computation.

That is incorrect. Lookup tables _are_ computations, and the algorithms
go by the name "memoisation", and are a form of optimisation where
space and time are traded.

[Bruno Marchal]

> On 19 Aug 2019, at 03:25, Russell Standish <li...@hpcoders.com.au> wrote:
>
> On Sat, Aug 17, 2019 at 12:17:38PM +0200, Bruno Marchal wrote:
>>
>> You cannot identify a computation and a representation of that computation. So
>> the answer is no: the blockhead or the infinite look-up table does not process
>> a computation.
>
> That is incorrect. Lookup tables _are_ computations, and the algorithms
> go by the name "memoisation", and are a form of optimisation where
> space and time are traded.

Only for finite functions. An infinite look-up table is not an algorithm or machine. Programs, machines, algorithm are required to be finitely given.

If we allow infinite look-up table, all functions from N to N becomes computable.

Bruno

>
>
> --
>
> ----------------------------------------------------------------------------
> Dr Russell Standish Phone 0425 253119 (mobile)
> Principal, High Performance Coders
> Visiting Senior Research Fellow hpc...@hpcoders.com.au
> Economics, Kingston University http://www.hpcoders.com.au
> ----------------------------------------------------------------------------
>

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[Jason Resch]

On Sat, Aug 17, 2019 at 5:17 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 16 Aug 2019, at 19:06, Jason Resch <jason...@gmail.com> wrote:

Would Chaitin's constant also qualify as a compact description of the universal dovetailing (though being a single real number, rather than a set of rational complex points)?

It does not. In fact Chaitin’s set (or “real number”) is not creative (Turing universal) but “simple", in that Post sense given above. You can’t compute anything with Chaitin’s number. It is like a box which contains all the gold of the universe, but there is no keys to open that box.

But “Post's number” , which decimals says if the nth program, in an enumeration of programs without inputs, stop or not, is creative, and “equivalent” with a UD* (seen properly in the right structure). But  the term ”compact” does not really apply here, unless perhaps you write the digits in smaller and smaller font so that you can write it all on one page.

You can look at Chaitin’s number as a maximal compression of Post’s number. Post number is deep, in Bennett sense, where Chaitin numbers is shallow and ultra-random. Chaitin’s number is the Post’s number with all the redundancies removed. You cannot do anything with it, except gives a non constructive proof of Gödel’s incompleteness (which was already in Emil Post, but without that “probability” interpretation of “simplicity”.

If Chatin's number is a maximally compressed Post's number, what makes one creative and the other simple, or one a representation of dovetailing and the other not?  Both require infinite computing resources dovetailing on all computations in order to generate them (don't they?).  I think I am missing something here.

 

An interesting paper, that Brent points to me some years ago, which shows this and more is the paper by Calude and Hay: "Every Computably Enumerable Random Real Is Provably Computably Enumerable Random" (arXiv:0808.2220v5). Here: https://arxiv.org/abs/0808.2220

From the abstract: "We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA"

This is surprising to me.  I thought it was generally easy to prove something is Turing universal, simply by programming it to match some other universal machine.  I will have to read it to see how.

 

That paper is also useful to see that PA can prove the existence of universal numbers, computations, … (without assuming anything in physics, which could help some people here). But it is a bit technical. It also assume that ZF is arithmetically sound, which I believe, but is not that obvious, especially with Mechanism!

Both Chaitin and Post numbers contains all the secrets (of the universal dovetailing), but Chaitin’s number, by removing all the redundancies, is unreadable, and just as good as total randomness or mess. Post’s numbers on the contrary is comprehensible by all universal machines, so to speak.

Put in another way: with Post numbers, there is full hope for a decent measure on the relative computational histories. With Chaitin’s number, there is no measure at all, like if all computational histories were unique, somehow.

I view Chatin's number as Post's number compressed so greatly you need to run a Busy_beaver(N) number of steps to decompress N bits.  Is this accurate?

 

This does not mean that Chaitin’s number is not interesting for Mechanism. I think it will play some role in the thermodynamic of the computationalist physical reality, but not in its origin (which Post’s number does).

Could you clarify the meaning of the thermodynamics of computationalist physical reality?  Is it equating physical randomness with the limit of the infinite complexity of the dovetailing computations?

Jason

 

[Bruno Marchal]

On 24 Aug 2019, at 00:23, Jason Resch <jason...@gmail.com> wrote:

On Sat, Aug 17, 2019 at 5:17 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 16 Aug 2019, at 19:06, Jason Resch <jason...@gmail.com> wrote:

Would Chaitin's constant also qualify as a compact description of the universal dovetailing (though being a single real number, rather than a set of rational complex points)?

It does not. In fact Chaitin’s set (or “real number”) is not creative (Turing universal) but “simple", in that Post sense given above. You can’t compute anything with Chaitin’s number. It is like a box which contains all the gold of the universe, but there is no keys to open that box.

But “Post's number” , which decimals says if the nth program, in an enumeration of programs without inputs, stop or not, is creative, and “equivalent” with a UD* (seen properly in the right structure). But  the term ”compact” does not really apply here, unless perhaps you write the digits in smaller and smaller font so that you can write it all on one page.

You can look at Chaitin’s number as a maximal compression of Post’s number. Post number is deep, in Bennett sense, where Chaitin numbers is shallow and ultra-random. Chaitin’s number is the Post’s number with all the redundancies removed. You cannot do anything with it, except gives a non constructive proof of Gödel’s incompleteness (which was already in Emil Post, but without that “probability” interpretation of “simplicity”.

If Chatin's number is a maximally compressed Post's number, what makes one creative and the other simple, or one a representation of dovetailing and the other not?  Both require infinite computing resources dovetailing on all computations in order to generate them (don't they?).  I think I am missing something here.

Chaitin’s number is the result of the computation, which consists in compressing the Post’s number, and making it totally unreadable. The result of a computation is not equivalent with the computation leading to that result. (That’s why the Adam machine which compute a very long time to get the answer 42 is funny :). It can be shown that such computations exist. Some predicate can be arbitrarily hard, despite they will answer 0, or 1. That can be proved by diagonalistation.

 

An interesting paper, that Brent points to me some years ago, which shows this and more is the paper by Calude and Hay: "Every Computably Enumerable Random Real Is Provably Computably Enumerable Random" (arXiv:0808.2220v5). Here: https://arxiv.org/abs/0808.2220

From the abstract: "We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA"

This is surprising to me.  I thought it was generally easy to prove something is Turing universal, simply by programming it to match some other universal machine.  I will have to read it to see how.

Even if this method works for some big, but finite sample of numbers, that would not be equivalent to proving that the machine does its work.

By Rice theorem, we have something more general. The set of number x such that phi_x computes the factorial function (say) is not computable. We cannot test if a program compute factorial or not. You can guess that such a program would solve all halting problems. Imagine the program search for a proof in ZF of the Riemann hypothesis, and if you find it output the usual factorial (n). If you decide algorithmically if that code computes the factorial function, you would be able to solve mechanically if Riemann hypothesis is true.

But you can also prove Rice theorem directly, using the second recursion theorem. If you can decide if a program compute some given function, you could use them to build a recursive permutation without fixed point, which cannot exist by the recursion theorem.

Most attributes of programs are non computable.

 

That paper is also useful to see that PA can prove the existence of universal numbers, computations, … (without assuming anything in physics, which could help some people here). But it is a bit technical. It also assume that ZF is arithmetically sound, which I believe, but is not that obvious, especially with Mechanism!

Both Chaitin and Post numbers contains all the secrets (of the universal dovetailing), but Chaitin’s number, by removing all the redundancies, is unreadable, and just as good as total randomness or mess. Post’s numbers on the contrary is comprehensible by all universal machines, so to speak.

Put in another way: with Post numbers, there is full hope for a decent measure on the relative computational histories. With Chaitin’s number, there is no measure at all, like if all computational histories were unique, somehow.

I view Chatin's number as Post's number compressed so greatly you need to run a Busy_beaver(N) number of steps to decompress N bits.  Is this accurate?

I think so. Chaitin numbers encodes not more than the Halting oracle, in a maximally compressed way.

 

This does not mean that Chaitin’s number is not interesting for Mechanism. I think it will play some role in the thermodynamic of the computationalist physical reality, but not in its origin (which Post’s number does).

Could you clarify the meaning of the thermodynamics of computationalist physical reality?  Is it equating physical randomness with the limit of the infinite complexity of the dovetailing computations?

That seems more like the arithmetical explanation of the quantum indeterminacy. The thermodynamics would be more related to some identification of the length of a finite computation and its code. A short code leading to a long computation would contain more energy than a short code leading to a short computation, up to some constant. But the length of the computation is not enough, and it is better to use the depth of it (following Bennett). The UD would have infinite “energy", like arithmetic, but the program “10 GOTO 10” despite leading to an infinite computation has basically no energy at all.

Bruno

Jason

 

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[Jason Resch]

On Sat, Aug 24, 2019 at 4:52 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 24 Aug 2019, at 00:23, Jason Resch <jason...@gmail.com> wrote:

On Sat, Aug 17, 2019 at 5:17 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 16 Aug 2019, at 19:06, Jason Resch <jason...@gmail.com> wrote:

Would Chaitin's constant also qualify as a compact description of the universal dovetailing (though being a single real number, rather than a set of rational complex points)?

It does not. In fact Chaitin’s set (or “real number”) is not creative (Turing universal) but “simple", in that Post sense given above. You can’t compute anything with Chaitin’s number. It is like a box which contains all the gold of the universe, but there is no keys to open that box.

But “Post's number” , which decimals says if the nth program, in an enumeration of programs without inputs, stop or not, is creative, and “equivalent” with a UD* (seen properly in the right structure). But  the term ”compact” does not really apply here, unless perhaps you write the digits in smaller and smaller font so that you can write it all on one page.

You can look at Chaitin’s number as a maximal compression of Post’s number. Post number is deep, in Bennett sense, where Chaitin numbers is shallow and ultra-random. Chaitin’s number is the Post’s number with all the redundancies removed. You cannot do anything with it, except gives a non constructive proof of Gödel’s incompleteness (which was already in Emil Post, but without that “probability” interpretation of “simplicity”.

If Chatin's number is a maximally compressed Post's number, what makes one creative and the other simple, or one a representation of dovetailing and the other not?  Both require infinite computing resources dovetailing on all computations in order to generate them (don't they?).  I think I am missing something here.

Chaitin’s number is the result of the computation, which consists in compressing the Post’s number, and making it totally unreadable. The result of a computation is not equivalent with the computation leading to that result. (That’s why the Adam machine which compute a very long time to get the answer 42 is funny :). It can be shown that such computations exist. Some predicate can be arbitrarily hard, despite they will answer 0, or 1. That can be proved by diagonalistation.

Perhaps 42 is the answer to life the universe and everything.  I computed that it requires 42 characters (bytes) of Chaitin's constant to encode the first Googol digits of Post's number: Log(10)/Log(2) * 100 / 8 = 41.524101125

That's less space than this sentence requires. ;-)

 

An interesting paper, that Brent points to me some years ago, which shows this and more is the paper by Calude and Hay: "Every Computably Enumerable Random Real Is Provably Computably Enumerable Random" (arXiv:0808.2220v5). Here: https://arxiv.org/abs/0808.2220

From the abstract: "We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA"

This is surprising to me.  I thought it was generally easy to prove something is Turing universal, simply by programming it to match some other universal machine.  I will have to read it to see how.

Even if this method works for some big, but finite sample of numbers, that would not be equivalent to proving that the machine does its work.

By Rice theorem, we have something more general. The set of number x such that phi_x computes the factorial function (say) is not computable. We cannot test if a program compute factorial or not. You can guess that such a program would solve all halting problems. Imagine the program search for a proof in ZF of the Riemann hypothesis, and if you find it output the usual factorial (n). If you decide algorithmically if that code computes the factorial function, you would be able to solve mechanically if Riemann hypothesis is true.

But you can also prove Rice theorem directly, using the second recursion theorem. If you can decide if a program compute some given function, you could use them to build a recursive permutation without fixed point, which cannot exist by the recursion theorem.

Most attributes of programs are non computable.

 

That paper is also useful to see that PA can prove the existence of universal numbers, computations, … (without assuming anything in physics, which could help some people here). But it is a bit technical. It also assume that ZF is arithmetically sound, which I believe, but is not that obvious, especially with Mechanism!

Both Chaitin and Post numbers contains all the secrets (of the universal dovetailing), but Chaitin’s number, by removing all the redundancies, is unreadable, and just as good as total randomness or mess. Post’s numbers on the contrary is comprehensible by all universal machines, so to speak.

Put in another way: with Post numbers, there is full hope for a decent measure on the relative computational histories. With Chaitin’s number, there is no measure at all, like if all computational histories were unique, somehow.

I view Chatin's number as Post's number compressed so greatly you need to run a Busy_beaver(N) number of steps to decompress N bits.  Is this accurate?

I think so. Chaitin numbers encodes not more than the Halting oracle, in a maximally compressed way.

 

This does not mean that Chaitin’s number is not interesting for Mechanism. I think it will play some role in the thermodynamic of the computationalist physical reality, but not in its origin (which Post’s number does).

Could you clarify the meaning of the thermodynamics of computationalist physical reality?  Is it equating physical randomness with the limit of the infinite complexity of the dovetailing computations?

That seems more like the arithmetical explanation of the quantum indeterminacy. The thermodynamics would be more related to some identification of the length of a finite computation and its code. A short code leading to a long computation would contain more energy than a short code leading to a short computation, up to some constant. But the length of the computation is not enough, and it is better to use the depth of it (following Bennett). The UD would have infinite “energy", like arithmetic, but the program “10 GOTO 10” despite leading to an infinite computation has basically no energy at all.

Do you envision there being any special properties (for instance, in the amplifications of the measure) of those programs shorter than the dovetailer, in terms of their contribution to to the statistics of the machine psychology?  Could you view those programs in a similar way to those patterns in the Mandelbrot seen between subsequent manifestations of the whole Mandelbrot set's shape? 

Jason

[Brent Meeker]

On 8/24/2019 1:35 PM, Jason Resch wrote:

That seems more like the arithmetical explanation of the quantum indeterminacy. The thermodynamics would be more related to some identification of the length of a finite computation and its code. A short code leading to a long computation would contain more energy than a short code leading to a short computation, up to some constant. But the length of the computation is not enough, and it is better to use the depth of it (following Bennett). The UD would have infinite “energy", like arithmetic, but the program “10 GOTO 10” despite leading to an infinite computation has basically no energy at all.

Do you envision there being any special properties (for instance, in the amplifications of the measure) of those programs shorter than the dovetailer, in terms of their contribution to to the statistics of the machine psychology?

I'd like to know what the "statistics" are.  Given all these values floating around there can be infinitely many statistics formed.

Brent